A British royal and a famed U.S. lawyer have each vigorously denied an anonymous claim that they were involved in the sex scandal that led to jail time for Jeffrey Epstein, a Florida financier who pleaded guilty to soliciting an underage prostitute.

The claims against Prince Andrew, the son of Queen Elizabeth II, and Alan Dershowitz, one of the most recognizable names in American law, surfaced this week in a civil lawsuit over a deal...